1-m P(y,t) = 3ty - y2 t ryct,

Ui

8F

K= 3y t p'(t)

8P

at= N

n”& N = 3y t ly'ct,

w&icrrHni N = 2tt3y,

1 iliua’a zt t3y = 3y t q'(t)

a&l v’(t) = 2t

yrct, = $"(t)dt

= s Ztdt : q'(t) = 2t

mn F(y,t) = 3ty - y= + pet,

F(y,tl = 3ty - y= •t ta : tP = qct,

F(y,t) = c

(3t - 2y)iiy t (3~ t Ztldt = 0

2tdy t ydt = 0

ah4M = 2t uia

at=2

aNN= y uia

Gt

2ytdy + y2dt = 0

nl5~li3~5~naurta~d~n5~ (integrating factor)

2 + u(t)y = w(t) : w(t) f 0

Go dy + (uy - w)dt = 0

Idy t I(uy - w)dt = 0

dsC M = I

uav N = I(uy-w)

Ga

1 a1-- = u

1 at

y(t) = Ae-lu(tldt

I (tl q *+t: Ll = u(t)

. .

G-S

F

J IL&&rF(y,t) = e dy + yct1

I udt= e J dy t q(t) : u I iiulii Y

.ruAt= ye t qct1

I UdLF(y,t) = ye t lyct1

aF IlA,itt =

w e t ly’ct,

BF

at= N

Iu,rN = yue t q'(t) ff

L ill”;Ui~.ru,t JYdC

e (uy-w) = yue t q'(t)

a&l q’(t) = eIu.t

(uy - WI - yueIt&cat

Iu*t I udt I u*t= UYe - we - yue

w-inshit

F(y,t) = ye t $fct,

F(y,t) = yeJudt - weJ IYdtdt

yes- _ weJ-dtdt = , cJ-sLz.t J sudt

y(t) = e cc+ we dt)

-Iory(t) = e

Iucatdt)

dy;TT:tuy = w

y(t) = e -slA.t (A +

I

SYdLwe dt)

luii aih-r&ihuun &I:

g+x, = t

iJEll u = 2t

uati w= t

a&i Judt = ta+k : k = d?&

y(t) = e-c2+*l CA t tect2+k'dt,s

-2 -L= e e (A t

ste?dtl

-2 --L= e e (A t e*

ste*'dtl

-tz 1 +*= Ae-*‘e-* + e (;e + cl

= Ae-ze-k + t. + e+‘c2

= (*e-k + c)e -i2 + $

: B = Ae-' + c

dyx+uy = w

(general solut

y(tl

ion) L U:J

-I= e

dYx t 4ty = 4t

Kim w = 4t

WEU Judt = 2ta : esdlRsYl

y(t) = e-ltS (A ts

4te'+ldt 1

-22CA t

I

4t a*'= e

4ted(2@)1

= e-" CA t e2'* d(2t2) 15

-2t2= e (A t ezZ) : nmfi1flsi= Ae-2tz f qe-2t~e't~

= A/" + 1 -2tz at': e e = 1.

y(t) = e-IYdt

(A t weJUdfdtlI

'I uif aunl4~~l#un ita:

$+a, = b

i& u = a

uw w = b

&I judt = at

y(t) = e-"(A tJbe"dt 1

Y(t) = e-” CA +. be” d(at) ,a

.

= Ae-” + ba

I Beneral solution

7-m y(t) = Ae-" + 6a

LiO t q 0: y(O) = A&b

a

= At!a

A = y(O) - ba

y(t) =b -mt

[y(O) - -1e t ba a

: definite solution

1 dl

rx= ps

I(t) HX?Biiij k37n58uan77a3nu ( r a t e o f i n v e s t m e n t f l o w )9

S WJ?fGs wwlh~h~lum5aa~ ( m a r g i n a l p r o p e n s i t y t o s a v e )

P W?fliT~ L57~~55nnwviay ( c a p a c i t y - c a p i t a l r a t i o )

t #~lEii5 L?al ( t i m e )

y(t) = e -I

y = I

u = -p(t)s(t)

W = 0

$udt = J-&'(t)s(t)dt

PS P(t,)S(ta)

PS

PSt

0 1311 0

J:p(t)s(t,dt

t t

cn) (Y)

P(t)s(tldt = abt'dt

1P(tIs(t)dt = ;abt3 I wd1aaR

I(i) = Ae$abt3

:

uasriio t = 0:

I(O) = Aa' = A

I(t) = I(Ole:abt3

: 5ll13a11ia&uI

1

Cdl(t)/dtl I(O)abt' e:abt

=I(t)

I(O) e:abt'

= abt'

= p(t)s(t) : p(t,)=at; s(tl=bt

4 . asll.

ALLEN, R.G.D. Mathematical Economnics. 2d ed., New York: St.Martin's

Press, Inc., 1959.

BAUMOL, W. J. Eccnomic Dynamics: An Introduction. 3d ed., New York:

The Macmillan Company, 1970.

CHIANG, A. C. Fundamenta f Methods of Mathematical Economics. 3d ed.,

New York: McGraw-Hi 11 Book Company, Inc., 1984.

CODDINGTON, E. A., AND N. LEVINSON. Theory of Ordinary Differential

Equations. New York: McGraw-Hill Book Company, 1955.

UOMAR, E.O. Essays in the Theory of Economic Growth. Fair Lawn, N.J.:

Oxford University Press, 1957.

FORD, L. R. D i f f e r e n t i a l E q u a t i o n s . New York; McGraw-Hill Book Com-

pany, 1933.

LEIGHTON, W. An Introduction to the Theory of Differential Equations.

New York: McGraw-Bill Book Company, 1952.

ROWEROFT, J. 8. Mathematical Economics : An Integrated Approach,

London; Paul Chapman Publishing, 1994.

SYDSAETER, K. , M a t h e m a t i c s f o r Economic A n a l y s i s .

N . J . : Prentice-Hall, Inc., 1994.

TAKAYANA, A. Analytical Methods in Economics.

Wheatsheaf, 1993.

Englewood Cliffs,

London: Harvester

YAMANE, T. Uathesatics for Economists: An Elemeritary Survey. Zd ed.,

Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1968.

.

titular i n t e g r a l : yP I W%JWii~BWi3LtWlhlJ ( g e n e r a l s o

Wat~~fl~WltiflT~ ( d e f i n i t e s o l u t i o n ) YoJ&.JnTTk!Jal$l& Cd*

e q u a t i o n ) FiO‘td~~

LPWW (par-

lution) bat!

i f f e r e n t i a l

(n)dy-+ 4y = 12d t

Lj;, y(Ol = 2

(ll)dy--zy= 0d t

l&l y(O) = 9

(6-l) 2 + 1oy = 15 tfa y(O) = 0

(91 2$+ 4y = 6 ifa y(o) = 1:

cn) dy-+ y= 4d t

rfa y(o) = 0

(ll)dyTic

= 23 kiia ~(0) = 1

. (Aldyt-

5y = 0 i”ua y(o) = 6

(3) 3gc 6y = 5 ria y(o) = 0.

cn) P& = 10 - 15P uav cl = 15P - 10 TdP

s flfl - = 5(Qd - Qs)dt

(11 Qe = 15P - 10dP

Uat! Q = 10 - 15Ps T@a - = 5(Q - (;I=)dt d

cm 2yt3dy t By't'dt = 0

(U) 3y'tdy t (y3 t 2t)dt = 0

cn, tC.1 t Zy)dy t y(l t y)dt = 0

dy(a), - t

2y4t t 3ta= 0

dt4y3t

cn, 2(t3 t l)dy t Bytdt = 0

(ll) 4y3tdy t (2y4 t 3t)dt = 0

(Ill f$sy = 15

(U)dydt + 2yt = O .

cc11 z t tay = 5ta rfa ~(01 = 6

cn) rfo p = tL URIS 8 = cl.3

(¶I) ria ,P = at* llm s = bt'

Qd = 35-3P

P = -45 + 5Ps

dP

dt= 0.7(Pd - Qsl